Question

Z-score and Percentile
Help please. Answer with explanation. Both circled numbers

Answer 1

I assume all these measures are normally distributed, in which case the median and mean are equal.

15a.
`P(X>66,650)=0.5` because the normal distribution is symmetric about the mean, and so 50% of families have an income greater than the average.

15b.
`P(X>178,500)=0.1` by definition of percentile. The 90th percentile is $178,500, so the top 10% of families earn more than this.

15c. Because it's continuous, this distribution has the property
`P(66,650<X<178,500)=P(X<178,500)-P(X<66,650)`. The top 10% earn more than $178,500, so the remaining 90% earn less. 50% of families earn more than the average, so 50% of families earn less than the average. Then
`P(66,650<X<178,500)=0.9-0.5=0.4` so 40% of families earn between the median and 90th percentile.

16a.
`P(X>570)=0.75` or 75% because the first quartile is $570, meaning 25% of renters pay less than this amount.

16b.
`P(X<708)=0.5` or 50% by the same reasoning used in (15a).

16c.
`P(570<X<708)=P(X<708)-P(X<570)=0.5-0.25=0.25` or 25% by the same reasoning as in (15c).